Pseudodifferential operators on a-modulation spaces

By Lasse Borup. 2004

59 citations recorded by the source corpus.

We study expansions of pseudodifferential operators from the Hormander class in a special family of functions called brushlets. We prove that such operators have a sparse representation in a brushlet system. Using this sparsity, we show that a pseudodifferential operator extends to a bounded operator between α -modulation spaces. These spaces were introduced by Grobner in [15]. They are, in some sense, intermediate spaces between the classical Besov and Modulation spaces.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6def790f9db4201de039a7af744ce4b0d1556e9f

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Modulation Spaces: Looking Back and Ahead (2006, methodology): “, [184, 119, 185, 151, 152, 169, 39, 129, 120, 2, 13, 19, 22, 186, 188, 187, 170, 12, 23].”
Generalized coorbit theory, Banach frames, and the relation to α‐modulation spaces (2008, result): “L. Borup, ‘Pseudodifferential operators on α-modulation spaces’, J. Funct.”
Banach frames for α-modulation spaces (2007, result): “Characterizations of α-modulation spaces by brushlet unconditional bases have been given by Borup and Nelson [5] and the mapping properties of pseudodifferential operators in Hörmander classes on α-modulation spaces have been studied by Holschneider and Nazaret [42] and Borup [4], as generalizations”
On anisotropic Triebel-Lizorkin type spaces, with applications to the study of pseudo-differential operators (2008, background): “The authors studied PDOs on α-modulation spaces in [3, 4].”
Time-Frequency Analysis of Sjöstrand's Class (2004, background): “This line of investigation and the emphasis on time-frequency techniques was continued in [11] [12] [23] [24] [31] [32].”
Ultradistributions and Time-Frequency Analysis (2006, background): “In particular, for ws(x, ξ) = (1 + |x|+ |ξ|)s, x, ξ ∈ R, s ≥ 0 properties of pseudodifferential operators whose symbols belong to the corresponding modulation spaces are given in [22], see also [15], [19], [29], [46], [47] for some generalizations.”
Continuous Frames, Function Spaces, and the Discretization Problem (2004, result): “For instance, the mapping properties of pseudodifferential operators in Hörmander classes on α-modulation spaces as investigated by Holschneider, Nazaret [42], and Borup [7] generalize classical results of Cordoba and Fefferman [10].”
Modulation Spaces on Locally Compact Abelian Groups (2003, background): “There is also work concerning pseudodifferential operators on modulation spaces by Bényi and Okoudjou [49], Hogan and Lakey [94], Labate [96], Pilipovic and Teofanov [104, 106, 105] and Toft [113], for example.”
Composition and spectral invariance of pseudodifferential Operators on Modulation Spaces (2006, background): “Labate [27] showed that M 1,1 M 1,1 ⊆ M 1,1 , and Toft [33] derived inequalities for the modulation spaces M p,1 in the style of Young's Theorem for convolution.”
MODULATION SPACES AND A CLASS OF BOUNDED MULTILINEAR PSEUDODIFFERENTIAL OPERATORS (2005, methodology): “Further developments using modulation spaces have since been obtained in [5], [15], [18], [21], [22], [25].”